First we point out that, similarly to the realization of \({\mathcal {H}}_n= {\mathcal {H}}(n, {\mathbb {R}})\) 
as a Hopf algebra of `moving frames', \(\Omega ^*((\mathop {\rm GL}\nolimits (n, {\mathbb {R}}))\)  has a natural representation
as a Hopf algebra of `moving coframes'. This representation is given by the coaction of
\(G =\mathop {\rm GL}\nolimits (n, {\mathbb {R}})\)  on the étale groupoid \(\Gamma _n= \mathop {\rm Diff}\nolimits _n \ltimes {\mathbb {R}}^n\)  determined by
the first jet map (analogous to the local coaction of \(\mathop {\rm GL}\nolimits (n, {\mathbb {R}})\)  on the \(C^*\) -algebra
of a codimension \(n\)  foliation, mentioned by Connes in [1]}). Specifically,
cf. [2]}, the formula
\( \rho (\alpha ) (\omega U_\varphi ) := J^*(\alpha )\omega U_\varphi , \quad \alpha \in \Omega ^*(G), \quad \omega \in \Omega _c^*(\Gamma _n) ,\) 
